Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions

TL;DR

This paper advances the understanding of smoothability in globally hyperbolic spacetimes, showing how to extend spacelike submanifolds to Cauchy hypersurfaces and constructing Cauchy temporal functions for spacetime splitting.

Contribution

It provides new results on extending spacelike submanifolds to Cauchy hypersurfaces and constructing Cauchy temporal functions, improving the understanding of spacetime structure and smoothability.

Findings

Any compact spacelike acausal submanifold with boundary can be extended to a Cauchy hypersurface.

A Cauchy temporal function can be constructed for any spacelike Cauchy hypersurface, enabling orthogonal splitting.

Continuous extensions are possible for achronal subsets, even when smooth extensions are not.

Abstract

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as $R \times S$ in a canonical way. Even more, accurate versions of this result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).